Two semigroup rings associated to a finite set of germs of meromorphic functions

TL;DR

This paper studies algebraic structures called semigroup rings derived from germs of meromorphic functions at a point, focusing on their properties and specific cases involving functions of finite order.

Contribution

It introduces and analyzes two semigroup rings associated with germs of meromorphic functions, especially in the context of functions of finite order and their holomorphic properties.

Findings

Characterization of the properties of the semigroup rings S^{hol} and ar S^{hol}

Detailed analysis of the case where  is the field of meromorphic functions of order <1

Results on the structure of these rings when functions have finitely many zeros and poles.

Abstract

We fix $z_0\in\mathbb C$ and a field $\mathbb F$ with $\mathbb C\subset \mathbb F \subset \mathcal M_{z_0}:=$ the field of germs of meromorphic functions at $z_0$. We fix $f_1,\ldots,f_r\in \mathcal M_{z_0}$ and we consider the $\mathbb F$-algebras $S:=\mathbb F[f_1,\ldots,f_r]$ and $\overline S:=\mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]$. We present the general properties of the semigroup rings \begin{align*} & S^{hol}:=\mathbb F[f^{\mathbf a}:=f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\ & \overline S^{hol}:=\mathbb F[f^{\mathbf a}:=f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0], \end{align*} and we tackle in detail the case in which $\mathbb F=\mathcal M_{<1}$ is the field of meromorphic functions of order $<1$ and $f_j$'s are meromorphic functions over $\mathbb C$ of finite order with a finite number of zeros and poles.

Equations

$$S^{hol}:=\mathbb{F}[f^{\mathbf{a}}:=f_1^{a_1}\cdots f_r^{a_r}:(a_1,\dots ,a_r)\in {\mathbb{N}}^r\text{ and }{f}^{\mathbf{a}}\text{ is holomorphic at }{z}_0],$$

$${\overline{S}}^{hol}:=\mathbb{F}[f^{\mathbf{a}}:=f_1^{a_1}\cdots f_r^{a_r}:(a_1,\dots ,a_r)\in {\mathbb{Z}}^r\text{ and }{f}^{\mathbf{a}}\text{ is holomorphic at }{z}_0],$$

$${\mathcal{O}}_0\cong \mathbb{C}\{z\}=\{\sum _{n=0}^{+\infty }{a}_n{z}^n:\frac{1}{{\mathrm{limsup}}_n\sqrt[n]{|a_n|}}>0\},$$

$$f(z)=\sum _{n=\ell }^{+\infty }{a}_n{z}^n,0<|z|<R,\text{ where }R=\frac{1}{\mathrm{limsup}\sqrt[n]{|a_n|}}>0\text{ and }\ell \in \mathbb{Z}.$$

$${\mathcal{M}}_0\cong Q(\mathbb{C}\{z\})=\mathbb{C}\{z\}[z^{-1}]\cong \frac{\mathbb{C}\{z\}[t]}{(1-zt)}.$$

$$S^{hol}:=\mathbb{F}[f^{\mathbf{a}}:=f_1^{a_1}\cdots f_r^{a_r}:{\mathrm{ord}}_{z=0}f(z)\ge 0,\mathbf{a}=(a_1,\dots ,a_r)\in {\mathbb{N}}^r]\subset S.$$

$${\overline{S}}^{hol}:=\mathbb{F}[f^{\mathbf{a}}:=f_1^{a_1}\cdots f_r^{a_r}:{\mathrm{ord}}_{z=0}f(z)\ge 0,\mathbf{a}=(a_1,\dots ,a_r)\in {\mathbb{Z}}^r]\subset \overline{S}.$$

$$S\cong \frac{\mathbb{F}[x_1,\dots ,x_r]}{\mathfrak{p}},$$

$$\overline{S}\cong \frac{\mathbb{F}[x_1^{\pm 1},\dots ,x_r^{\pm 1}]}{\mathfrak{q}}\cong \frac{\mathbb{F}[x_1,\dots ,x_r,y_1,\dots ,y_r]}{\overline{\mathfrak{p}}},$$

$$\mathfrak{q}\cong \frac{\overline{\mathfrak{p}}}{(x_1{y}_1-1,\dots ,x_r{y}_r-1)}.$$

$$S^{hol}:=\mathbb{F}[f^{\mathbf{a}}:=f_1^{a_1}\cdots f_r^{a_r}:\mathbf{a}=(a_1,\dots ,a_r)\in {\mathbb{N}}^r\text{ and }{\mathrm{ord}}_{z=0}{f}^{\mathbf{a}}(z)\ge 0]\subset S,$$

$${\overline{S}}^{hol}:=\mathbb{F}[f^{\mathbf{a}}:=f_1^{a_1}\cdots f_r^{a_r}:\mathbf{a}=(a_1,\dots ,a_r)\in {\mathbb{Z}}^r\text{ and }{\mathrm{ord}}_{z=0}{f}^{\mathbf{a}}(z)\ge 0]\subset \overline{S}.$$

$$S^{hol}:=\mathbb{F}[f^{\mathbf{a}}:a_1{\ell }_1+\cdots +a_r{\ell }_r\ge 0,\mathbf{a}\in {\mathbb{N}}^r].$$

$${\overline{S}}^{hol}:=\mathbb{F}[f^{\mathbf{a}}:a_1{\ell }_1+\cdots +a_r{\ell }_r\ge 0,\mathbf{a}\in {\mathbb{Z}}^r].$$

$$H:=\{\mathbf{a}=(a_1,\dots ,a_r):a_1{\ell }_1+\cdots +a_r{\ell }_r\ge 0\}\subset {\mathbb{N}}^r,$$

$$\overline{H}:=\{\mathbf{a}=(a_1,\dots ,a_r):a_1{\ell }_1+\cdots +a_r{\ell }_r\ge 0\}\subset {\mathbb{Z}}^r,$$

$$\mathbb{F}[H]:=\mathbb{F}[x^{\mathbf{a}}=x_1^{a_1}\cdots x_r^{a_r}:\mathbf{a}\in H]\subset \mathbb{F}[x_1,\dots ,x_r].$$

$$\mathbb{F}[\overline{H}]:=\mathbb{F}[x^{\mathbf{a}}=x_1^{a_1}\cdots x_r^{a_r}:\mathbf{a}\in \overline{H}]\subset \mathbb{F}[x_1^{\pm 1},\dots ,x_r^{\pm 1}].$$

$$\tilde{H}:=\{(\mathbf{b},\mathbf{c})\in {\mathbb{N}}^r\times {\mathbb{N}}^r:\mathbf{b}-\mathbf{c}\in \overline{H}\}.$$

$$\mathbb{F}[\overline{H}]\cong \frac{\mathbb{F}[\tilde{H}]}{(x_1{y}_1-1,\dots ,x_r{y}_r-1)}.$$

$$S^{hol}\cong \frac{\mathfrak{p}+\mathbb{F}[H]}{\mathfrak{p}}\cong \frac{\mathbb{F}[H]}{\mathfrak{p}\cap \mathbb{F}[H]}.$$

$${\overline{S}}^{hol}\cong \frac{\mathfrak{q}+\mathbb{F}[\overline{H}]}{\mathfrak{q}}\cong \frac{\overline{\mathfrak{p}}+\mathbb{F}[\tilde{H}]}{\overline{\mathfrak{p}}}\cong \frac{\mathbb{F}[\tilde{H}]}{\overline{\mathfrak{p}}\cap \mathbb{F}[\tilde{H}]}.$$

$$(i){\ell }_1>0,\dots ,{\ell }_p>0,{\ell }_{p+1}=\cdots ={\ell }_r=0,\text{ where }p\ge 0$$

$$(ii){\ell }_1<0,\dots ,{\ell }_q<0,{\ell }_{q+1}=\cdots ={\ell }_r=0,\text{ where }q\ge 1$$

$$(iii){\ell }_1>0,\dots ,{\ell }_p>0,{\ell }_{p+1}<0,\dots ,{\ell }_q<0,{\ell }_{q+1}=\cdots ={\ell }_r=0,1\le p<q\le r.$$

$$\mathbb{F}[H]=\mathbb{F}[x_1,\dots ,x_r]\text{ and }\mathbb{F}[\overline{H}]=\mathbb{F}[x_1,\dots ,x_p,x_{p+1}^{\pm 1},\dots ,x_r^{\pm 1}].$$

$$\mathbb{F}[H]=\mathbb{F}[x_{q+1},\dots ,x_r]\text{ and }\mathbb{F}[\overline{H}]=\mathbb{F}[x_1^{-1},\dots ,x_q^{-1},x_{q+1}^{\pm 1},\dots ,x_r^{\pm 1}].$$

$$\Phi :\mathbb{F}[t_1,\dots ,t_m]\to \mathbb{F}[H],\Phi (t_j):=v_j,1\le j\le m.$$

$$S^{hol}\cong \frac{\mathbb{F}[t_1,\dots ,t_m]}{{\Phi }^{-1}(\mathfrak{p}\cap \mathbb{F}[H])}.$$

$$\tilde{\Phi }:\mathbb{F}[t_1,\dots ,t_s]\to \mathbb{F}[\tilde{H}],\tilde{\Phi }(t_j)=w_j,1\le j\le s.$$

$${\overline{S}}^{hol}\cong \frac{\mathbb{F}[t_1,\dots ,t_s]}{{\tilde{\Phi }}^{-1}(\overline{\mathfrak{p}}\cap \mathbb{F}[\tilde{H}])}.$$

$$L_t=|\{\mathrm{supp}(v):v\in \mathbb{F}[H],|\mathrm{supp}(v)|=t\}|\text{ and }{N}_t=\left(\genfrac{}{}{0ex}{}{r}{t}\right)-\left(\genfrac{}{}{0ex}{}{r-2}{t-1}\right)+1.$$

Full text

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and we tackle in detail the case in which $\mathbb{F}=\mathcal{M}_{<1}$ is the field of meromorphic functions of order $<1$ and $f_{j}$’s are meromorphic functions over $\mathbb{C}$ of finite order with a finite number of zeros and poles.

2010 MSC: 30D30; 30D20; 16S36.

Keywords: meromorphic functions, entire functions, semigroup rings.

1 Introduction

Let $z_{0}\in\mathbb{C}$ and let $g$ be a holomorphic function at $z_{0}$, that is $g$ is holomorphic on an open domain $U\subset\mathbb{C}$ with $z_{0}\in U$. Replacing $g(z)$ with $g(z-z_{0})$, we can assume that $z_{0}=0$. Given two holomorphic functions $g_{1}$ and $g_{2}$ at [math] we say that $g_{1}\sim g_{2}$ if there exist an open domain $U\ni z_{0}$ such that $g_{1}|_{U}=g_{2}|_{U}$. $\sim$ is an equivalence relation. A class of equivalence of $\sim$ is called a germ of holomorphic function. We denote $\mathcal{O}_{0}$ the ring of germs of holomorphic functions at [math]. It is well known that

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the ring of convergent power series, which is an one dimensional local regular ring with the maximal ideal $\mathfrak{m}=z\mathbb{C}\{z\}$.

Let $f$ be a meromorphic function at [math], that is there exists an open domain $U\subset\mathbb{C}$, $0\in U$, and two holomorphic functions $g,h:U\rightarrow\mathbb{C}$ such that $f(z)=\frac{g(z)}{h(z)}$ for all $z\in U\setminus\{0\}$. It is well known that $f$ has a Laurent expansion

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If $a_{\ell}\neq 0$, the number $\operatorname{ord}_{z=0}f(z):=\ell$ is called the order of zero of $f$ at [math]. If $\ell\geq 0$, then $f$ is holomorphic at [math] and has a zero of order $\ell$ at [math]. If $\ell<0$, then [math] is a pole of order $-\ell$ of $f$. As in the holomorphic case, we define $\mathcal{M}_{0}$ the ring of germs of meromorphic function at [math]. We have that $\mathcal{M}_{0}$ is the quotient field of $\mathcal{O}_{0}$ and hence

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In order to simplify the notation, we denote by $f$ a holomorphic (meromorphic) function at [math] and its germ.

We fix a field $\mathbb{F}$ such that $\mathbb{C}\subset\mathbb{F}\subsetneq\mathcal{M}_{0}$ and some germs $f_{1},\ldots,f_{r}\in\mathcal{M}_{0}$. We consider the $\mathbb{F}$-algebras $S:=\mathbb{F}[f_{1},\ldots,f_{r}]$ and $\overline{S}:=\mathbb{F}[f_{1}^{\pm 1},\ldots,f_{r}^{\pm 1}]$. Our aim is to study the $\mathbb{F}$-subalgebras

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In the second section, we present the general properties of $S^{hol}$ and $\overline{S}^{hol}$, using the methods from [5]. Theorem $2.3$, Theorem $2.4$ and Theorem $2.5$ are simple generalizations of the main results from [5], hence we omit the proofs.

In the third section, we present our main results of the paper. We let $\mathbb{F}:=\mathcal{M}_{<1}$ be the field of meromorphic functions of order $<1$ and we let $f_{1},\ldots,f_{r}$ be some meromorphic functions of finite order with finite number of zeros and poles. In Proposition $3.1$ we prove that such functions are of the form $R(z)e^{P(z)}$, where $R(z)$ is a rational function and $P(z)$ is a polynomial. In Theorem $3.3$ we prove that if $P_{1},P_{2},\ldots,P_{r}\in\mathbb{C}[z]$ are polynomials such that $P_{j}-P_{k}$ are non-constant for all $j\neq k$, then the functions $f_{j}(z)=e^{P_{j}(z)}$, $1\leq j\leq r$, are linearly independent over $\mathbb{F}$. Moreover, if $d_{j}:=\deg(P_{j})\geq 1$ for $1\leq j\leq r$ and $d_{j}\neq d_{k}$ for all $j\neq k$, then $f_{1},\ldots,f_{r}$ are algebraically independent over $\mathbb{F}$. In Corollary $3.4$ we prove similar conclusions, when we replace $f_{j}$’s with linear combinations $h_{j}=\sum_{k=1}^{r}g_{jk}f_{k}$, $1\leq j\leq r$, where $g_{jk}\in\mathbb{F}$ and the determinant $\det((g_{jk})_{j,k})$ is nonzero. In Corollary $3.5$ we prove that if $\varphi\in\mathbb{F}$ and $f_{j}(z)=e^{P_{j}(z)}$, $1\leq j\leq r$, are as in the hypothesis of Theorem $3.3$, then $\varphi,f_{1},\ldots,f_{r}$ are linearly (algebraically) independent over $\mathbb{C}$. We conclude our paper with Example $3.6$.

2 Preliminaries

Let $\mathcal{M}_{0}$ be the field of germs of meromorphic functions at [math]. Let $\mathbb{F}$ be a field such that $\mathbb{C}\subset\mathbb{F}\subsetneq\mathcal{M}_{0}$ and let $f_{1},\ldots,f_{r}\in\mathcal{M}_{0}$. Let $S:=\mathbb{F}[f_{1},\ldots,f_{r}]$ and $\overline{S}:=\mathbb{F}[f_{1}^{\pm 1},\ldots,f_{r}^{\pm 1}]$. Since $S$ is a domain, we have

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where $\mathfrak{p}\subset\mathbb{F}[x_{1},\ldots,x_{r}]$ is a prime ideal. Similarly,

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where $\mathfrak{q}\subset\mathbb{F}[x_{1}^{\pm 1},\ldots,x_{r}^{\pm 1}]$ is a prime ideal and $\overline{\mathfrak{p}}\subset\mathbb{F}[x_{1},\ldots,x_{r},y_{1},\ldots,y_{r}]$ is a prime ideal such that

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We consider the $\mathbb{F}$-subalgebras

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Let $\ell_{j}:=\operatorname{ord}_{z=0}f_{j}(z)$, for $1\leq j\leq r$. From (2.3) it follows that

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Similarly, from (2.4) it follows that

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We consider the semigroups

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and their associated toric ring

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We consider the semigroup

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with its associated toric ring $\mathbb{F}[\widetilde{H}]$. One can easily check that

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From (2.1), (2.5) and (2.9) it follows that

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From (2.2), (2.6), (2.10) and (2.12) it follows that

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There are three cases to consider:

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In the case $(i)$, we have that

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In the case $(ii)$, we have that

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Assume we are in the case $(iii)$. Let $v_{1},\ldots,v_{m}\in\mathbb{F}[x_{1},\ldots,x_{r}]$ be the minimal monomial set of generators of the $\mathbb{F}$-algebra $\mathbb{F}[H]$. In [5, Proposition 1.3(1)] we proved that $m\geq r$. We consider the natural epimorphism

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$I_{H}:=\operatorname{Ker}(\Phi)$ is called the toric ideal of $H$, see [9] for further details. From (2.13) and (2.15) it follows that

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Now, assume that $w_{1},\ldots,w_{s}$ are the minimal monomial generators of the $\mathbb{F}$-algebra $\mathbb{F}[\widetilde{H}]$. We consider the natural epimorphism

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The ideal $I_{\widetilde{H}}:=\operatorname{Ker}(\widetilde{\Phi})$ is the toric ideal of $\widetilde{H}$. From (2.14) and (2.17) it follows that

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Remark 2.1**.**

Let $K/\mathbb{Q}$ be a finite Galois extension. For the character $\chi$ of the Galois group $G:=Gal(K/\mathbb{Q})$ on a finite dimensional complex vector space, let $L(s,\chi):=L(s,\chi,K/\mathbb{Q})$ be the corresponding Artin L-function ([3, P.296]). Artin conjectured that $L(s,\chi)$ is holomorphic in $\mathbb{C}\setminus\{1\}$ and $s=1$ is a simple pole. Brauer [4] proved that $L(s,\chi)$ is meromorphic in $\mathbb{C}$, of order $1$. Let $\chi_{1},\chi_{2},\ldots,\chi_{r}$ be the irreducible characters of $G$. Let $f_{1}:=L(s,\chi_{1}),\ldots,f_{r}:=L(s,\chi_{r})$.***

Artin [2, Satz 5, P. 106] proved that $f_{1},\ldots,f_{r}$ are multiplicatively independent. F. Nicolae proved in [7] that $f_{1},\ldots,f_{r}$ are algebraically independent over $\mathbb{C}$. This result was extended in [6] to the field $\mathcal{M}_{<1}$ of meromorphic functions of order $<1$. Let $\mathbb{F}$ be a field such that $\mathbb{C}\subset\mathbb{F}\subset\mathcal{M}_{<1}$. We consider $S:=\mathbb{F}[f_{1},\ldots,f_{r}]$, $\overline{S}:=\mathbb{F}[f_{1}^{\pm 1},\ldots,f_{r}^{\pm 1}]$, $S^{hol}$, $\overline{S}^{hol}$, $H$ and $\overline{H}$ as above. An extensive study of the semigroup rings $\mathbb{F}[H]\cong S^{hol}$ and $\mathbb{F}[\overline{H}]\cong\overline{S}^{hol}$ was done in [5], in the frame of Artin L-functions.***

We recall the several results from [5], which hold in our (more general) context.

Theorem 2.2**.**

([5, Proposition 1.3(2), Theorem 1.4, Proposition 2.2]) In the case $(iii)$, the following are equivalent:

1. (1)

$I_{H}=(0)$. 2. (2)

$\mathbb{F}[H]$* is minimally generated by $r$ monomials.* 3. (3)

$\ell_{1}>0$, $\ell_{2}<0,\ldots,\ell_{q}<0$, $\ell_{q+1}=\cdots=\ell_{r}=0$, where $q\geq 2$, and $\ell_{1}|\ell_{j}$ for $2\leq j\leq q$. 4. (4)

$\mathbb{F}[H]=\mathbb{F}[x_{1},x_{1}^{m_{2}}x_{2},\ldots,x_{1}^{m_{q}}x_{q},x_{q+1},\ldots,x_{r}]$, where $m_{j}=-\frac{\ell_{j}}{\ell_{1}}$, $2\leq j\leq q$. 5. (5)

$\mathbb{F}[\overline{H}]=\mathbb{F}[x_{1},(x_{1}^{m_{2}}x_{2})^{\pm 1},\ldots,(x_{1}^{m_{q}}x_{q})^{\pm 1},x_{q+1}^{\pm 1},\ldots,x_{r}^{\pm 1}]$, where $m_{j}=-\frac{\ell_{j}}{\ell_{1}}$, $2\leq j\leq q$.

Given a monomial $v\in\mathbb{F}[x_{1},\ldots,x_{r}]$, the support of $v$ is the set $\operatorname{supp}(v)=\{x_{j}\;:\;x_{j}|v\}$. For $1\leq t\leq r-1$, we consider the numbers:

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Theorem 2.3**.**

([5, Theorem 1.6]) Except the case $(ii)$, the following are equivalent:

1. (1)

$\mathbb{F}[H]=\mathbb{F}[x_{1},\ldots,x_{r}]$. 2. (2)

$I_{H}=(0)$* and there exists $1\leq t\leq r-1$ such that $L_{t}\geq N_{t}$.*

Theorem 2.4**.**

([5, Theorem 1.13, Proposition 2.3]) In the case (iii), if $\ell_{1}=\cdots=\ell_{p}=1,\;\ell_{p+1}=\cdots=\ell_{q}=-1\text{ and }\ell_{q+1}=\cdots=\ell_{r}=0$, then we have:

1. (1)

$\mathbb{F}[H]=\mathbb{F}[x_{1},\ldots,x_{p},x_{q+1},\ldots,x_{r},x_{j}x_{k}\;:\;1\leq j\leq p,\;p+1\leq k\leq q]$. 2. (2)

$\mathbb{F}[\overline{H}]=\mathbb{F}[x_{1},\ldots,x_{p},x_{q+1}^{\pm 1},\ldots,x_{r}^{\pm 1},(x_{j}x_{k})^{\pm 1}:1\leq j\leq p,\;p+1\leq k\leq q].$** 3. (3)

Letting $\Phi:\mathbb{F}[t_{1},\ldots,t_{p},t_{q+1},\ldots,t_{r},t_{jk}\;:\;1\leq j\leq p,\;p+1\leq k\leq q]\to\mathbb{F}[H]$, $\Phi(t_{j})=x_{j}$, $\Phi(t_{jk})=x_{j}x_{k}$, we have:

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3 Main results

We denote $\mathcal{O}$ the domain of entire functions. We have that

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Let $f\in\mathcal{O}$. If there exist a positive number $\rho$ and constants $A,B>0$ such that

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then we say that f has an order of growth $\leq\rho$. We define the order of growth of $f$ as

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For each integer $k\geq 0$ we define canonical factors by

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Let $f\in\mathcal{O}$ be an entire function with the order of growth $\rho$. From Hadamard’s Theorem (see for instance [8, Theorem 5.1]), it follows that

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where $k=\lfloor\rho\rfloor$, $z_{1},z_{2},\ldots$ are the non-zero zeros of $f$, $P$ is a polynomial of degree $\leq k$ and $m$ is the order of the zero of $f$ at $z=0$. In particular, if the number of zeros of $f$ is finite, then

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It is well known that the field of meromorphic functions on $\mathbb{C}$, denoted by $\mathcal{M}$ is the quotient field of $\mathcal{O}$. Moreover, if $f$ is meromorphic with order of growth $\leq\rho$, then $f$ is the quotient of two holomorphic functions with order of growth $\leq\rho$. For any $\rho>0$, we denote $\mathcal{O}_{<\rho}$ the domain of entire functions with order of growth $<\rho$, and $\mathcal{M}_{<\rho}$ the quotient field of $\mathcal{O}_{<\rho}$, that is the field of meromorphic functions of order $<\rho$.

Proposition 3.1**.**

If $f$ is a meromorphic function with order of growth $\rho$ with finitely many zeros and poles, then $\rho$ is an integer and $f(z)=R(z)e^{P(z)}$, where $R(z)\in\mathbb{C}(z)$ is a rational function, and $P\in\mathbb{C}[z]$ is a polynomial of degree $\rho$.

Proof.

Since $f$ is meromorphic of order $\rho$, we can write $f(z)=\frac{g(z)}{h(z)}$, where $g$ and $h$ are holomorphic of order $\leq\rho$ and at least one of then has the order of growth $\rho$. From (3.3) it follows that $\rho$ is integer and

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where $P_{1},Q_{1},P_{2}$ and $Q_{2}$ are polynomials with $\max\{\deg(P_{1}),\deg(P_{2})\}=\rho$. Therefore

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Since $f$ has the order of growth $\rho$, it follows that $\deg(P_{1}-P_{2})=\rho$, as required. ∎

Remark 3.2**.**

Let $f_{1},\ldots,f_{r}$ be some meromorphic functions with finite orders of growth $\rho_{1},\ldots,\rho_{r}$ and finitely many zeros and poles. From Proposition $3.1$ it follows that

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where $R_{j}\in\mathbb{C}(z)$ and $P_{j}\in\mathbb{C}[z]$ for $1\leq j\leq r$. We have the $\mathbb{C}(z)$-algebra isomorphisms

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Since $\mathbb{C}(z)$ is a subfield of $\mathcal{M}_{<1}$, it follws that we have the $\mathcal{M}_{<1}$-algebra isomorphism

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However, in general $S^{hol}\subsetneq(\mathcal{M}_{<1}[e^{P_{1}(z)},\ldots,e^{P_{r}(z)}])^{hol}=\mathcal{M}_{<1}[e^{P_{1}(z)},\ldots,e^{P_{r}(z)}]$, as the functions $f_{1},\ldots,f_{r}$ could have poles at $z=0$.

In the following theorem, we give a criterion for the linear (algebraic) independence of the functions $f_{j}(z)=e^{P_{j}(z)}$, $1\leq j\leq r$, over the field $\mathcal{M}_{<1}$.

Theorem 3.3**.**

Let $P_{1},P_{2},\ldots,P_{r}\in\mathbb{C}[z]$ be polynomials such that $P_{j}-P_{k}$ is non-constant for any $j\neq k$. Let $d_{j}:=\deg(P_{j})$ for $1\leq j\leq r$ and

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1. (1)

The holomorphic functions $f_{1},\ldots,f_{r}$ are linearly independent over $\mathcal{M}_{<1}$. 2. (2)

If $d_{j}\geq 1$ for all $1\leq j\leq r$ and $|\{d_{1},\ldots,d_{r}\}|=r$ then $f_{1},\ldots,f_{r}$ are algebraically independent over $\mathcal{M}_{<1}$.

Proof.

(1) Note that $f_{j}$ is an entire functions of order $d_{j}:=\deg(P_{j})$, for any $1\leq j\leq r$. We use induction on $r\geq 1$. The case $r=1$ is obvious. Assume $r\geq 2$ and let $g_{1},\ldots,g_{r}\in\mathcal{M}_{<1}$ such that

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If $g_{r}=0$, then we are done by induction hypothesis. Without any loss of generality, we can assume that $g_{r}$ is identically $1$. It follows that

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Differentiating (3.4) it follows that

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Since $(P_{j}-P_{r})-(P_{k}-P_{r})=P_{j}-P_{k}$ are non-constant for all $1\leq j\neq k\leq r-1$, by induction hypothesis, it follows that $g^{\prime}_{j}+(P_{j}-P_{r})g_{j}=0$, for all $1\leq j\leq r-1$, hence

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If $C_{j}\neq 0$, since $\deg(P_{r}-P_{j})\geq 1$, from (3.5) it follows that $g_{j}$ is a holomorphic function of order $\geq 1$, a contradiction. Hence $g_{j}=0$ for all $1\leq j\leq r-1$ and thus we get a contradiction from (3.4).

(2) Let $Q\in\mathbb{F}[t_{1},\ldots,t_{r}]$ be a polynomial such that $Q(f_{1},\ldots,f_{r})=0$. We have that

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and only a finite number of $g_{\mathbf{a}}$’s are nonzero. Hence

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For any $\mathbf{a}\in\mathbb{N}^{r}$, we have

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Let $\mathbf{b}\neq\mathbf{a}\in\mathbb{N}^{r}$. Since the $d_{j}$’s are pairwise disjoint, the polynomial

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in non-constant. From (3.7), (3.8) and (i) it follows that the set $\{f^{\mathbf{a}}\;:\;\mathbf{a}\in\mathbb{N}^{r}\}$ is linearly independent over $\mathbb{F}$. Hence, from (3.6), we get $Q=0$, as required. ∎

Corollary 3.4**.**

Let $g_{jk}\in\mathcal{M}_{<1}$ for all $1\leq j,k\leq r$ such that

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where $A\subset\mathbb{C}$ is a non-discrete subset. In the hypothesis of Theorem $3.3$, the meromorphic functions $h_{j}:=\sum_{k=1}^{r}g_{jk}f_{k}$, $1\leq j\leq k$, are linearly independent over $\mathcal{M}_{<1}$.

Moreover, in the hypothesis $(2)$ of Theorem $3.3$, the functions $h_{1},\ldots,h_{r}$ are algebraically independent over $\mathcal{M}_{<1}$.

Proof.

As $D(z)$ is non constant on a non-discrete subset $A\subset\mathbb{C}$, it follows that $D\in\mathcal{M}_{<1}$ is nonzero. Since, from Theorem $3.3$, $\{f_{1},\ldots,f_{r}\}$ are linearly independent over $\mathcal{M}_{<1}$, it follows that $\{h_{1},\ldots,h_{r}\}$ are also linearly independent over $\mathcal{M}_{<1}$.

Since $D$ is nonzero and $f_{1},\ldots,f_{r}$ are algebraically independent over $\mathbb{F}$, it follows that the map

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is a $\mathcal{M}_{<1}$-algebra isomorphism, hence $h_{1},\ldots,h_{r}$ are algebraically independent. ∎

Corollary 3.5**.**

Let $\varphi\in\mathcal{M}_{<1}$ be a a non-constant function and let $P_{1},P_{2},\ldots,P_{r}\in\mathbb{C}[z]$ be non-constant polynomials of degrees $d_{1},d_{2},\ldots,d_{r}$ such that $P_{j}-P_{k}$ is non-constant for any $j\neq k$. Let $f_{j}(z)=e^{P_{j}(z)}$ for $1\leq j\leq r$. Then:

1. (1)

$\varphi,f_{1},\ldots,f_{r}$* are linearly independent over $\mathbb{C}$.* 2. (2)

If $|\{d_{1},\ldots,d_{r}\}|=r$ then $\varphi,f_{1},\ldots,f_{r}$ are algebraically independent over $\mathbb{C}$.

Moreover, if $A=(a_{ij})_{0\leq i,j\leq r}$ is a nonsingular matrix with entries in $\mathbb{C}$, and

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then the conclusions $(1)$ and $(2)$ holds if we replace $\varphi,f_{1},\ldots,f_{r}$ with $g_{0},g_{1},\ldots,g_{r}$.

Proof.

(1) We consider a linear combination

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If $a=0$, then $a_{1}=\cdots=a_{r}=0$ from Theorem $3.3(1)$. Assume $a\neq 0$. Note that, at most one of the polynomials $P_{j}$’s is constant. We consider two cases:

(i) If $d_{1}=0$ and $d_{j}\geq 1$ for $2\leq j\leq r$, then $f_{1}\in\mathbb{C}$. Let

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From (3.9) it follows that

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According to the proof of Theorem $3.3(1)$, this yields a contradiction.

(ii) If $d_{j}\geq 1$ for $1\leq j\leq r$, then we let

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From (3.9) it follows that

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which, according to the proof of Theorem $3.3(1)$, yields a contradiction.

(2) From Theorem $3.3(2)$ it follows that $f_{1},\ldots,f_{r}$ are algebraically independent over $\mathcal{M}_{<0}$ and hence over $\mathbb{C}(\varphi)$. On the other hand, the nonconstant function $\varphi$ is algebraically independent over $\mathbb{C}$. Thus $\varphi,f_{1},\ldots,f_{r}$ are algebraically independent over $\mathbb{C}$.

The last assertion follows from Corollary $3.4$. ∎

We conclude our paper with a list of examples.

Example 3.6**.**

(1) Let $f_{1}(z)=z,f_{2}(z)=\frac{e^{-z}}{z^{2}},f_{3}(z)=e^{z}$ and $S=\mathbb{C}[f_{1},f_{2},f_{3}]$. According to Corollary $3.5$, the meromorphic functions $f_{1},f_{2},f_{3}$ are linearly independent over $\mathbb{C}$. One can easily check that*

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On the other hand, we have that

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We consider the semigroups

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Since $\ell_{1}=\operatorname{ord}_{z=0}f_{1}=1$, $\ell_{2}=\operatorname{ord}_{z=0}f_{2}=-2$ and $\ell_{3}=\operatorname{ord}_{z=0}f_{3}=0$, from Theorem $2.2$ it follows that

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From (2.13) and (2.16) it follows that

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From (2.14) and (2.18) it follows that $\overline{S}^{hol}=\mathbb{C}[z,e^{-z},e^{z}]\cong\mathbb{C}[t_{1},t_{2}^{\pm 1}]$.**

(2) Let $f_{1}(z)=z,f_{2}(z)=\sin z,f_{3}(z)=\frac{e^{z}}{z},\;f_{4}(z)=\frac{e^{-z}}{z}$ and $S=\mathbb{C}[f_{1},f_{2},f_{3},f_{4}]$. We have*

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Since $\ell_{1}=\ell_{2}=1$ and $\ell_{3}=\ell_{4}-1$, from Theorem $2.3$ it follows that

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From (2.16) it follows that

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